Systematically convergent weak-coupling expansion of similarity-renormalized Hamiltonians at the bound-state formation scale in asymptotically free theories

The similarity renormalization group (SRG) for Hamiltonians replaces a singular canonical Hamiltonian by a family of band-diagonal effective Hamiltonians of width $\lambda$, constructed by a continuous unitary flow, with the aim of computing bound states of asymptotically free theories perturbatively down to the binding scale. Głazek and Wilson [Phys. Rev. D 57, 3558 (1998)] verified in an exactly solvable matrix model that the expansion in the running coupling reproduces the bound-state eigenvalue to about 10% at second order down to the bound-state formation scale, but becomes erratic and non-convergent beyond second order. The paper posed the problem of remedying this convergence failure — by resummation, by an explicitly identified irrelevant operator $\widetilde{h}(\lambda)$ to be renormalized coherently with the running coupling $\widetilde{g}$, by constituent-quark-model-type leading Hamiltonians, or by generalized flow generators — together with the dense-discretization (near-neighbour coupling) question, numerical round-off control, and the transfer of the validated strategy to QCD. Later work resolved the model-level convergence question via generator optimization (about 1% accuracy through sixth order in the same model), and the descendant RGPEP program realized the effective-gluon-mass idea at second order for heavy-flavor QCD. A precise open core nevertheless survives: no convergent higher-order (at least fourth order) similarity-RG Hamiltonian for QCD with simultaneous running coupling and effective masses has been constructed, and the $\widetilde{h}$-operator route, the many-states-per-scale question, and the constituent-picture interpretation of deep-inelastic sum rules remain untreated. The problem is open.

Background

Quantum chromodynamics (QCD), the theory of quarks and gluons, is asymptotically free: its coupling becomes small at short distances, so high-energy scattering is accessible to perturbation theory, while hadrons form as bound states at scales where the coupling is of order one. A long-standing goal is to compute hadron masses and wave functions directly from the QCD Hamiltonian in Minkowski space-time, which would also yield the boost-invariant wave functions that the parton model (the description of a fast hadron as a collection of quasi-free constituents carrying momentum fractions) uses in hard-scattering processes. The light-front, or front form, of Hamiltonian dynamics is attractive for this purpose because boosts act kinematically there, but the canonical front-form QCD Hamiltonian is a singular operator whose matrix elements between states of very different free energies diverge.

The similarity renormalization group (SRG) for Hamiltonians, introduced by Głazek and Wilson in the light-front program of Wilson et al., replaces the canonical Hamiltonian by a one-parameter family of effective Hamiltonians $H(\lambda)$ obtained through a continuous unitary transformation that suppresses matrix elements between states whose free energies differ by much more than a width $\lambda$, leaving a nearly band-diagonal matrix. A special case is Wegner's flow equation, whose generator is the commutator of the diagonal part of $H$ with $H$ itself. As $\lambda$ is lowered, the effective coupling grows like the running coupling (the scale-dependent effective strength of the interaction) of the asymptotically free theory; the hope is that $H(\lambda)$ can still be calculated in weak-coupling perturbation theory down to $\lambda$ of order the bound-state binding energy, where only a small window of effective states dominates the bound state and can be diagonalized numerically.

Głazek and Wilson tested precisely this in a $38\times 38$ matrix model, obtained by discretizing the two-dimensional Schrödinger equation with a delta-function potential — a quantum-mechanical model that exhibits asymptotic freedom and one bound state and whose exact similarity flow and spectrum are computable. They found that (i) inserting the Gaussian similarity factor and the exact running coupling, second-order perturbation theory yields the bound-state eigenvalue to about 10% accuracy down to $\lambda$ of order the bound-state formation scale (the width at which the bound-state eigenvalue appears on the diagonal of $H(\lambda)$, where the coupling is of order one); (ii) expansion in the initial bare coupling fails badly; and (iii) beyond second order in the running coupling the expansion turns erratic — the third- and fourth-order terms carry large alternating coefficients, successive terms nearly cancel, and accuracy is destroyed rather than improved. The constituent quark model (the phenomenology in which light hadrons are described as weakly interacting massive constituent quarks) enters as the suggested form of a better leading-order Hamiltonian whose difference from the true effective Hamiltonian might remain small enough to be computed perturbatively.

The paper closes by posing the problem of remedying this convergence failure and identifying what must be verified before the strategy can be transferred to QCD. The direct descendant of this program is the renormalization group procedure for effective particles (RGPEP), in which the similarity factors become momentum-space form factors of effective particles and a gluon mass parameter helps control the small-$x$ divergences (singularities at vanishing longitudinal momentum fraction) of light-front gluon dynamics; its current state is reviewed by Głazek in 2015 and 2026. Glueballs (bound states of gluons) and deep-inelastic-scattering sum rules (conservation laws constraining parton distributions) enter as QCD-specific questions about the constituent picture created by the similarity transformation.

Problem Statement

Can the weak-coupling expansion of similarity-renormalized Hamiltonians be made systematically convergent down to the bound-state formation scale in asymptotically free theories, and thereby deliver controlled Hamiltonian calculations of bound states — ultimately in QCD? Partial progress exists at the model level — an optimized generator makes the expansion systematic there — but that result alone does not close this objective, whose surviving open core is the QCD-level construction together with the still-untreated items among the linked open questions below. As posed by Głazek and Wilson (1998), this single objective decomposes into the following linked open questions, all posed quantitatively against the exactly solvable matrix model and its intended QCD transfer:

1. Remedy of the convergence failure beyond second order in the running coupling near the binding scale: determine quantitatively whether resummation, inclusion of the identified irrelevant operator $\widetilde{h}(\lambda)$ (an irrelevant operator, in the renormalization-group sense, is one whose coupling fades as the flow scale is raised; here $\widetilde{h}\sim\widetilde{g}^{2}$, and it is to be renormalized coherently with the running coupling $\widetilde{g}$), or improved leading-order Hamiltonians — a constituent-quark-model-type Hamiltonian whose difference from the true effective Hamiltonian is treated perturbatively — restores order-by-order improvement, and how much accuracy can be gained for narrow (small-$\lambda$) effective Hamiltonians.

2. Generalized flow generators (the class of flow equations generalizing Wegner's, whose generators reshape the rate and pattern at which off-diagonal matrix elements are eliminated): establish whether they accelerate or stabilize the perturbative expansion, and by what choice.

3. Dense discretization: determine the behavior of the method when the energy-scale spacing parameter $b$ is reduced toward 1, so that many states occur per energy scale — in particular, how large the effective energy range of near-neighbour interactions, which the similarity transformation does not attenuate, can become, and whether the narrow-window picture survives.

4. Numerical stability: control the round-off errors that arise when high-order expansion coefficients nearly cancel.

5. Extension to QCD: construct and validate the effective Hamiltonian at the orders required to include running couplings and effective (gluon-mass-type) terms simultaneously — at least fourth order in the weak-coupling expansion — and confront the constituent-picture questions: dynamical generation of effective gluon masses at small coupling, the interpretation of deep-inelastic-scattering sum rules in the constituent picture produced by the similarity transformation, and the effect of strong gluonic binding (glueballs) on the presence of constituent gluons in quark bound states.

References

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